## Factor Trail Game

• Lesson
3-5
1

When students play the Factor Trail game, they have to identify the factors of a number to earn points. Built into this game is cooperative learning — students check one another's work before points are awarded. The score sheet used for this game provides a built-in assessment tool that teachers can use to check their students' understanding.

In this lesson, students play the Factor Trail game in which they identify the factors of various numbers. Although practicing a mundane skill, students enjoy the work in this game because of the game scenario.

Explain to students that they will be playing a game involving factoring. Ask them what it means to factor a number, and then ask them to help you find all the factors of a number. You may want to choose a number with a lot of factors (24, 36, 60), or you can roll two dice or spin two spinners to generate the digits of a number randomly. However you choose a number, use the think-pair-share strategy to allow the class to identify the factors: first, give students one minute to "think" individually and come up with some factors of the number; then, give them another minute to discuss their lists with a partner; and finally, record the entire list of factors on the board or overhead projector via a class discussion.

You may then wish to give students a few more numbers to practice on their own before playing the game. Use these additional warm-up problems to determine how well students are able to factor numbers. Then, when students begin to play the game for practice, spend additional time with students who had difficulty.

After the warm-up, distribute the Factor Trail Game to all students. Note that two students can share the game board and rules that appear on the first and second pages, but all students will need their own score sheet.

 Factor Trail Game

The Factor Trail Game is a game for two players. Players move around the game board, landing on numbered squares. When landing on a square with a number, students should list all of the factors of that number on their score sheet. When a student believes that she has listed all of the factors, her opponent checks the list. If her oppponent identifies any factors not on the player's list, or if the opponent identifies any number on the player's list that is not a factor of the number, the opponent receives 10 points for identifying the error. (If the opponent notices multiple errors, 10 points are earned for each error.) If the player made no errors, however, then she receives points for that turn equal to the sum of the factors of the number.

Example: A player lands on 18. On her score sheet, she lists 1, 2, 3, 4, 6, 9, and 18 as factors. Upon indicating that she has completed her list, her opponent points out that 4 is not a factor of 18. Consequently, the opponent receives 10 points for identifying the error. On the other hand, if she had not included 4 on her list, she would have correctly identified all of the factors and received 1 + 2 + 3 + 6 + 9 + 18 = 39 points.

The game can be played with or without calculators. The use of calculators does not greatly influence the game, as students must still understand the concept and skill of factoring to be successful. However, if a secondary objective of the lesson is to have students practice mental arithmetic, then calculators should not be used.

As students play the game, circulate to offer assistance where necessary. This may involve settling a dispute between two students, or it may require intervening when you notice that students are making mistakes not caught by their opponents.

With a few minutes left in class, you may wish to pause all games and conduct a brief discussion using the questions that appear in the Questions For Students section below.

Assessments

1. Collect the score sheets of all students. The score sheets can be used to determine if students were correctly finding the factors of numbers. One of the benefits of using the score sheet occurs when a 0 is entered in the "Points Earned" column. This indicates that the student made a mistake when finding the factors of that number, so it is easier to identify areas of difficulty.

Extensions

1. Change the numbers on the game board. Note that all of the numbers are less than 100. For a more advanced game, include numbers in the hundreds or thousands.
2. Students can use their score sheets as the beginning of a list of abundant, deficient, and perfect numbers.
• A number is said to be abundant if the sum of its proper divisors is greater than the number itself.
• A number is said to be deficient if the sum of its proper divisors is less than the number itself.
• A number is said to be perfect if the sum of its proper divisors is equal to the number itself.

All numbers can be categorized as either abundant, deficient, or perfect. The mathematician Leonhard Euler identified the sigma function σ(n), which gives the sum of the positive divisors of n. Using this notation, then abundant numbers are those with σ(n) > 2n, deficient numbers are those with σ(n) < 2n, and perfect numbers are those with σ(n) = 2n.

Students can use the Abundant, Deficient, Perfect activity sheet to keep track of the numbers 1–100.

 Abundant, Deficient, Perfect Activity Sheet

Students can also research the fascinating life of Euler. One of the topics that Euler studied was amicable pairs (an amicable pair consists of two integers for which the sum of proper divisors of one number equals the other number, and vice versa). Because amicable pairs also depend on the factor sum of numbers, they are closely related to abundant, deficient, and perfect numbers.

Questions for Students

1. Which number on the game board has the most factors?
[The numbers 60, 72, 90 and 96 all have 12 factors.]
2. For which number on the trail will a player earn the most points?
[The sum of the factors of 96 is 252, offering the most points of any number on the board.]
3. In general, how many points are earned for a prime number?
[For a prime number p, then p + 1 points are earned. That is, the number of points is one more than the number itself, since the only factors of a prime number are the number and 1. For instance, 18 points are earned for the prime number 17, whereas 41 points are earned for the prime number 41.]

Teacher Reflection

• Did students enjoy the game?
• While students played the game, were they able to focus on the mathematics of the activities?
• What common errors did students make while finding the factors of a number? What misconceptions might have led to those errors, and how might those misconceptions be corrected?

### Learning Objectives

Students will:

• Practice identifying the integer factors of numbers up to 100.